Absolute Value absolute value - the distance from a number to 0 on the number line

Transcription

1 Introduction to Integers (positive & negative whole numbers) Natural (counting numbers): Whole numbers: Integers: Some Examples of Integers: Positive Negative Number Line Graphs Inequalities Ex a Write the correct relationship for each pair: Absolute Value absolute value - the distance from a number to 0 on the number line

2 Adding Integers Like signs - add amounts, keep same sign Different signs (2 integers) subtract amounts, choose sign of larger amount Adding More Than 2 Numbers - 2 methods Method 1: Add pairs from left to right, one at a time. Method 2: Add all positives, then all negatives separately. Combine the positive and negative totals Additive Inverse (also called opposite) has same amount (absolute value), opposite sign

3 15 Applications Ex A book shelf is supported by 2 blocks. The forces exerted are as shown. Add the forces. What does the result mean? Ex Trey s bank account begins with $348. He withdraws $160, deposits $457, then withdraws $300. 1) Write an expression using signed numbers to represent these transactions. 2) What is his final balance?

4 Subtracting Integers - same as adding the opposite of a number Subtracting Procedure 1. Change the subtraction sign to addition 2. Change the sign of the number subtracted 3. Add as before Ex Observe: Subtracting a negative has the same result as adding a positive Solving Equations Recall: When you move a term to the other side, reverse the sign or operation (inverse) Applications Financial/Business: N = R C C = cost (money spent) R = revenue (money received) N = net Profit occurs when the net is Loss occurs when the net is Ex A start-up business spends $7000 to purchase supplies and $1000/month in rent and utilities. The first year, $40,000 in merchandise is sold. What is the net profit or loss?

5 Multiplying & Dividing Integers; Exponents, Square Roots, Solving Recall: When no operator is shown is implied. Rules for Multiplying Integers: same sign: opposite signs: Ex Why is negative x positive = negative? Why is negative x negative = positive? More than 2 negative signs multiplied or divided: even number of negative signs: odd number of negative signs: Evaluating Exponential Form (-5) 2 = = (-2) 3 = =

6 18 Rules for Dividing Integers: same sign: opposite signs: Solving use inverse operation (related equation) Square Root - the number which, squared, gives the final number (also, the length of a square with a given area) ( x ) 2 = 25 Every positive number has 2 square roots Words: square roots of 81 Symbols: 81 When a square root is written as a symbol, it has only one root, positive or negative, but not both. Ex

7 19 Square root of a negative number: 36 Some perfect squares (helpful to memorize) Root number Square Applications Ex A car loan is taken out, and the amount to be paid back (including interest) is $36,000. If the term of the loan is 6 years, how much is repaid each month. Write amounts as signed numbers.

8 Order of Operations with Integers (including negative numbers) PEMDAS rules still apply. Absolute value bars and square root symbols are considered, which should be done. Be care of ( ) signs, especially with exponents. Recall: ( 4) 2 = 4 2 =

9 Additional Applications & Problem Solving Net/Revenue/Cost Formula: Voltage/Current/Resistance Formula: V = i R, where V = voltage (in volts) i = current (in amperes or amps) R = resistance (in ohms or ) Distance/Rate/Time Formula: d = r t

10 Translating & Evaluating Expressions Equation Has equal sign Process: Examples: Expression No equal sign Processes: Examples: Translating Phrases to Expressions We represent real quantities as variables, because these quantities keep changing. 4 important words: sum difference product quotient Examples (more in book) Addition the sum of x and 2 x plus y 5 more than y 3 added to some number 10 increased by a number Subtraction the difference of x and 2 x minus y 5 less than y 3 subtracted from some number 10 decreased by a number Note: Addition is commutative; subtraction is not. Multiplication the product of 2 numbers x times 3 x multiplied by 3 Division the quotient of 2 numbers x divided by 3 x divided into 3 twice some number triple a number Note: Multiplication is commutative; division is not.

11 23 Exponents x squared the square of x a number cubed the cube of a number Roots the square root of a number the cube root of a number y to the 4 th power p raised to the 7 th power Evaluating Expresions - find a number value 1. Replace each variable with a number. If it helps, you can replace each variable with parentheses first, then insert the number into the parentheses. 2. Calculate the final number using the correct order of operations. Note: Find the value of the expression has the same meaning as Evaluate

12 3.2 Introduction to Monomials; Combining Like Terms monomial (term) a number, variable, or product of numbers and variables (which may be raised to whole-number exponents). They contain NO Examples of monomials: 24 Examples of things that are NOT monomials: A monomial has 2 parts: 1. - the number part, including the sign 2. - the letters, including exponents Ex a Find the coefficient and variable part of each term below: Term Coefficient Variable Part Degree of Term Note: A term always has. It does not always have Degree of a term - the sum of the exponents of the variables for a single term Polynomials with One Variable most common Polynomial has 1 or more terms added or subtracted. Examples of polynomials: Binomial - has Trinomial has Descending Order - A polynomial written with highest powers first. Degree of a polynomial the degree of the highest power term only (there may be several terms) Like terms - terms that have exactly the same variable parts (including exponents) Ex Are the following pairs of terms like or unlike?

13 Adding and Subtracting Polynomials Simplest Form an expression with the fewest number of terms and symbols get rid of parentheses combine all like terms Polynomials in 1 variable (compare terms of polynomial to digits in arithmetic) Adding Polynomials Horizontal Method - Gather like terms in descending order, then combine Vertical Method - Stack like terms in descending order, then combine Perimeter - add the lengths of the sides Subtracting Polynomials - Change to an addition problem by 1. Changing subtract to add 2. Changing the sign of ALL terms in 2 nd parentheses (subtracted terms)

14 Exponent Rules; Multiplying Polynomials Product Rule of Exponents Ex a n a n b = n a+b Power to a Power Rule of Exponents (n a ) b = n ab Ex b Multiplying Monomials 1. Gather numbers, then like variables together 2. Multiply coefficients, then each type of variable separately. Raising a Monomial to a Power 1. Raise the coefficient to the power and evaluate 2. Raise each variable to the power and simplify Multiplying a Polynomial by a Monomial 1. Use the distributive law to multiply each term inside the parentheses 2. Simplify as above.

15 27 Multiplying 2 Polynomials: Book's Method 1. Multiply first term in first polynomial by second polynomial 2. Multiply each of the other terms by second polynomial 3. Gather like terms Multiplying 2 Polynomials : Box Method 1. Make a box, with each polynomial on one side 2. Multiply terms, filling each box 3. Gather like terms Multiplying 2 Binomials special case of 2 polynomials FOIL - First, Outer, Inner, Last Ex Conjugates - 2 binomials that have the same 2 terms, but the second terms have opposite signs. Rule: the product of conjugates is the difference of squares

16 Prime Numbers and Greatest Common Factors (GCF s) Prime Number a natural number greater than 1 whose only factors are 1 and itself. (a number that can t be broken down to smaller factors) Note: All even numbers, except 2, are NOT prime, since is a factor. Composite Number a natural number that has factors other than 1 or itself. Every composite number CAN be broken down to smaller factors, and eventually to prime factors. Finding ALL the factors of a number Ex Find all the factors of 60. How do we know when to stop? Deciding if a number is prime or composite - test each factor to see if there are any smaller factors Finding the Prime Factorization of a Number 1. Break down the number into its smallest (prime) factors 2. Write the factors as a product, using exponents if needed

17 29 Finding the Greatest Common Factor by Lists 1. List all factors of each number 2. Find the largest factor common to both lists Finding the GCF by factoring 1. Write each number it its prime factored form, including exponents 2. Choose common bases only ; choose the lowest power common to both numbers 3. Multiply

18 Exponents Rules; Dividing Polynomials/Introduction to Factoring Quotient Rule of Exponents - When dividing like bases, subtract exponents Zero Rule of Exponents any number raised to the zero power = Dividing Monomials 1. Group coefficients and similar variables together 2. Divide using exponent rules (subtract exponents) or cancellation Dividing Polynomials by a Monomial 1. Separate terms to divide each term by the monomial 2. Simplify as before Finding an Unknown Factor Long way: Divide the product by the known factor Short way: Eyeball the product to see what s missing

19 Additional Applications and Problem Solving Guidelines for Word Problems 1. Decide which formula is appropriate. Write it. 2. Substitute the known numbers or quantities into the variables of the formula. 3. Solve for the unknown variable. Perimeter, Area, Volume formulas P = sum of all sides (any shape) P = 2L + 2W (rectangle) A = LW (rectangle) A = bh (parallelogram) V = LWH Ex a Given the shape with dimensions shown, find Surface Area SA = 2LW + 2WH + 2HL Falling Objects h = -16t 2 + h o h = current height, t = time, h o = original height

20 Net/Cost/Revenue - Formula: Ex Amy buys jewelry-making tools for $100. Materials cost $10 for earrings, $15 for bracelets, and $20 for pendants. She charges $20 for earrings, $30 for bracelets, and $35 for pendants. 1) Write an equation describing cost, where E = # of sets of earrings, B = # of bracelets, and P = # pendants sold. 32 2) Write an equation describing the revenue 3) Write an equation describing the net 4) What is the net if she sells 5 sets of earrings, 1 bracelet and 3 pendants? Is it a profit or loss? 5) Suppose she fills an order next week for 3 sets of earrings, 4 bracelets, and 2 pendants. If she can reuse the same tools (reducing costs), what is her net?

21 Equations and Solutions Recall the contrast between equations and expressions: Equation Has Process: Looks like: Expression No Process(es): Looks like Solutions of an Equation Solve - get a number that makes the equation true Solution - a number that makes the equation true Checking to see if a number is a solution - Procedure 1. Replace the variable(s) with the number 2. Simplify and determine if the equation is true

22 The Addition/Subtraction Property of Equality Linear Equation an equation where each variable term has degree 1 linear : non-linear: Addition/Subtraction Property of Equality - you can add or subtract the same amount from both sides without changing the solution A = B is equivalent to A C = B C or A + C = B + C Goal in Solving: Get the variable by itself on one side of equation (isolate the variable); get the number on the other side Solving Procedure for equations with 1 complication 1. Combine like terms on the left side of the = sign; then combine like terms on the right 2. Look at the variable. Decide what is junk. 3. Decide how the junk is connected to the variable. Do the reverse to undo the junk. 4. Isolate the variable 5. Check Solving with variables on 2 sides of equation get rid of smaller variable term first

23 The Multiplication/Division Property of Equality Multiplication/Division Property of Equality - you can multiply or divide by the same amount on both sides without changing the solution A = B is equivalent to AC = BC or A/C = B/C Undo junk by doing the reverse operation Solving with both added/subtracted & multiplied/divided junk Solving Procedure (summary) 1. Simplify if needed (clear parentheses, combine like terms) 2. Get all variable terms on one side, by adding or subtracting from both sides; combine into one variable term 3. Get rid of added/subtracted junk 4. Get rid of multiplied/divided junk

24 Translating Word Sentences to Equations Recall 4 important words: sum, difference, product, quotient Recap Examples Addition the sum of a number and 2 5 more than a number 3 added to t a number increased by 10 Subtraction the difference of 3 and a number 3 less than y 7 subtracted from a number x decreased by 8 Multiplication the product of 4 and a number 6 times p half of x twice a number Division the quotient of a number and 11 4 divided by a number $15 per 5 gallons Equal

25 Applications and Problem Solving Procedure for solving 1. Let one quantity be an expression built around x, and the other be x. 2. Assign quantities to the expression. 3. Make an equation using the expressions 4. Solve and ANSWER THE QUESTION 5. Check (optional) Triangle and Angle Definitions 1. Equilateral triangle triangle that has 2. Isosceles triangle triangle that has 3. Supplementary angles angles that add up to 4. Complementary angles angles that add up to 5. Congruent angles angles with Solving for 1 type using a formula Money Formula: value number = amount

26 38 Common constructions (algebraic expressions) for 2 types 1. one side is 4 ft more than the other side 2. there are twice as many children as adults 3. there are 10 coins total Solving for 2 types using a table category value number amount Ex category value number amount

27 Fractions, Mixed Numbers, Rational Expressions Fraction part out of a whole part whole = numerator denominator Examples of fractions: Proper fraction fraction where the numerator is less than the denominator Improper fraction - fraction where the numerator is greater than or equal to the denominator. Mixed number an integer combined with a (proper) fraction Which is better? In general, improper fractions are better for mixed numbers are better for Converting Improper Fractions to Mixed Numbers 1. Divide numerator by denominator using long division 2. Write quotient as whole number part, write remainder over the denominator for the proper fraction part.

28 40 Converting Mixed Numbers to Improper Fractions 1. Multiply the whole number part by the denominator 2. Add the numerator to the product 3. Put the sum over the denominator Rational number a number that can be written as a/b Fraction a number that actually is written as a/b Rational numbers include: all all all Graphing a fraction on a number line Equivalent Fractions Reducing and Building Up reducing - cancel common factors building up multiply by the missing factor Comparing Fractions using > < or = To compare fractions it s helpful to have the same denominators

29 5.2 Simplifying Fractions and Rational Expressions lowest terms a fraction whose numerator and denominator have no common factors 41 equivalent fractions fractions that have the same value when reduced to lowest terms Reducing to Lowest Terms (2 methods) Method 1 Divide numerator and denominator by common factors (or the GCF) until no common factors remain Method 2 Write prime factors and cancel Reducing Improper Fractions 1. Cancel common factors (required) 2. Convert to a mixed number (optional)

30 42 Acceptable Reduced Forms: Acceptable Not Acceptable Not all answers are required to be shown in mixed number form. Convention: If the problem begins with improper fractions, give the final answer as an improper fraction (same is true of mixed numbers), unless the final answer is asked for in a different form. Reducing Expresions with Variables Caution: Cannot cancel added terms (do not confuse addition with multiplied factors, which can be canceled). 2 5x 2 5x 2 5x x 2 Compare to:

31 5.3 Multiplying Fractions, Mixed Numbers, Rational Expressions Sign of a Fraction - A fraction is considered a single number and should have only one sign and only one opposite; for example the opposite of 5 3 is Acceptable form for proper and improper fractions: Acceptable form for mixed numbers: Translating Expressions & Equations - Of" usually means multiply, not 43 Multiplying Procedure 1. Reduce by cancelling any numerator with any denominator until all common factors are gone. 2. Multiply 2 numerators and put in the numerator 3. Multiply 2 denominators and put in the denominator Multiplying Mixed Numbers 1. Convert to an improper fraction 2. Cancel common factors 3. Multiply 4. Convert back to a mixed number for final answer (what you started with)

32 Estimating Products (operation is ) book says round to nearest half, but I ll accept nearest whole number, unless the rounded number is 0 1. Round each mixed number to the nearest whole number. In the fraction part, look at denominator number what is half the denominator? If the numerator is smaller than half the denominator, the fraction is < ½ (round down) If the numerator greater than or equal to half the denom., the fraction is > ½ (round up) 2. If there is no integer part and the fraction is less than ½, keep the fraction as is, since factors in products should not be rounded to Multiply the numbers 44 Multiplying Rational Expressions (polynomials) - cancel variables as well as numbers Fractions Raised to a Power raise each factor in numerator & denominator separately Applications Area of a Triangle: A = ½ bh

33 45 Ex 2/3 of all students in the Central Valley graduate from high school. If 1/6 of HS graduates earn a BA/BS degree, what fraction of all CV students earn BA/BS degrees? Ex If ¾ of the stores in a mall are open for business, and there are 300 stores: 1) How many stores are open? 2) How many are closed? 3) What fraction are closed? 4) If 1/3 of the closed stores are being remodeled, what fraction of all stores are being remodeled? 5) How many stores are being remodeled?

34 Dividing Fractions, Mixed Numbers, Rational Expressions reciprocals 2 numbers whose product is 1; the flip or upside down of a fraction Dividing Fractions Procedure 1. Keep first fraction the same. Invert the second fraction and change division to multiplication (invert and multiply). 2. Cancel any common factors 3. Multiply numerators and denominators 4. Simplify Dividing Mixed Numbers 1. Convert the mixed numbers to improper fractions 2. Divide as above 3. Convert back to a mixed number Complex fractions (more than 2 layers) - can be written as a division problem

35 47 Dividing Rational Expressions (variables) Square Roots of Fractions Procedure 1. Simplify fraction if possible 2. Separate numerator and denominator 3. Find the square roots separately Solving Fractional Equations (multiplied/divided junk only) Method 1: Divide both sides by the fraction Method 2: Multiply both sides by the reciprocal Applications

36 Least Common Multiples multiple the product of a whole number and the original number (a multiple s size is the same as or than the original number) multiples of 6: 6, 12, multiples of 8: Common multiples: Least common multiple smallest multiple that is common to both (or all) numbers Finding the LCM by listing 1. Write lists of the multiples of each number 2. Compare the lists until the first common multiple is found LCM Shortcuts 1. If there all numbers have no common factors, the LCM is the product 2. If one number is a perfect multiple of all the numbers, the big number is the LCM. Finding the LCM by Prime Factorization 1. Find the prime factorization of every number 2. Choose each unique base 3. Write largest exponent for each base 4. Multiply the numbers to get the LCM Finding the Greatest Common Factor 1. Find the prime factorization 2. Choose only bases common to all the numbers 3. Write smallest exponent for each base 4. Multiply the numbers to get the GCF Factor - a broken down piece, same as or smaller than the number Multiple a multiplied product, same as or larger than the number

37 49 Sometimes we want to force fractions to have the same denominator Building an equivalent fraction to an LCD Procedure 1. Find the LCD. Write a new fraction with the LCD and an empty (unknown) numerator 2. Find the missing factor needed to change the old denominator into the new denominator. If you can t eyeball it, divide the new denominator by the old. 3. Multiply both numerator and denominator by the missing factor to make the new fractions

38 5.6 Adding and Subtracting Fractions, Mixed Numbers, and Rational Expressions Two or more fractions must have the same denominator to be added (e.g. same size of pizza slices) 50 Adding/Subtracting with Same Denominator (Procedure) 1. Add or subtract numerators write in numerator 2. Keep same denominator 3. Reduce if needed Caution: Adding/Subtracting with Different Denominators (Procedure) 1. Find the LCD 2. Build all fractions to the LCD 3. Add/subtract numerators, keep same denominator 4. Simplify Reminder: LCDs are only needed for adding/subtracting fractions. Multiplying fractions does not require having the same denominators. Adding/Subtracting Mixed Numbers Method 1: Convert to improper fractions, combine, then convert back to mixed numbers

39 51 Method 2: Combine whole number and fraction parts separately. Borrow or carry if needed Estimating Sums (operation is ) rounding to zero IS acceptable 1. Round to the nearest whole number - up if fraction, down if fraction 2. Combine numbers Solving fractional equations (added/subtracted junk only) - get rid of junk by doing the reverse process

40 5.7 Order of Operations - Evaluating & Simplifying Fractions & Rat. Expressions Recall order of operations: PEMDAS For multiplying improper fractions are best For adding either mixed numbers or improper fractions can be used Ex a 52 Evaluate substitute a number for each variable Area of a Trapezoid A = ½ h(a+b) or A = ½(a+b)h Note: ½(a+b) is the average of a and b Adding/Subtracting Polynomials gather like terms together, then add/subtract

41 Solving Equations both multiplied/divided AND added/subtracted junk Solving Procedure different from simplifying expressions 1. Find LCD 2. Multiply both sides by LCD to GET RID of fractions (do NOT build up to keep fractions) 3. Solve as before Translating sentences to equations Of Applications Ex Ray s scores are 87, 52, and 76. What must he score on the next test for a C average?

42 Decimals and Rational Numbers Converting decimals to fractions or mixed numbers 1. Write the numbers to the left of the point as a whole number 2. Write the digits to the right as numerator 3. Write the place value in the denominator 4. Reduce if needed Trick: you can tack extra zeroes onto the end if needed Word Names decimal words are similar to the fraction they represent 1. Number left of decimal point: 2. Point: 3. Number right of point: Graphing on a number line Ex Ex

43 55 Comparing Decimals using > or < 1. If needed, tack on zeroes to make the decimals equal length 2. Compare digits beginning starting immediately after the point (if needed, tack on zeroes to make numbers the same length) 3. When digits differ, the larger number gives the larger amount Rounding 1. Look at the specified digit, then look at the next place value (immediately after the specified one) 2. If the next digit is 0 4, keep the desired digit 5 9, round up Ex Round to the nearest a) thousandth b) hundredth c) whole number (unit) d) ten e) hundred Ex Round to the nearest a) hundredth b) tenth c) whole number (unit) d) ten e) hundred

44 Adding and Subtracting Decimals Adding Procedure 1. Stack numbers so place values align (line up decimal points) 2. Add the numbers keeping decimal point in the same place (If it helps, tack on zeroes to make the same number of places in each decimal) Subtracting Procedure 1. Stack the numbers, with greater abs. value (amount) on top. 2. Subtract digits, tacking on zeroes if necessary 3. Determine the final sign of the number Adding/Subtracting Polynomials 1. Put like terms together 2. Combine numbers for each type

45 57 Solving (added/subtracted junk only) Isolate the variable Application Ex Find the final balance Transaction Credits Debits Balance $ Pay check PG & E Rent Costco Refund Note: There is no transaction in the line before the first balance.

46 Multiplying Decimals; Exponents Multiplying decimals gives the same value as multiplying equivalent fractions Multiplying Procedure 1. Stack the 2 numbers, one on top of the other. You don t need to line up decimal points. 2. Multiply the numbers as if they were whole numbers. 3. Count the number of places to the right of all decimals. 4. Move the decimal point to the left that number of places 5. Fill in zeroes if necessary Multiplying/Dividing Shortcuts powers of 10 Multiplying by a power of 10 makes a number (move the decimal point to the right the same # of places as the power). Dividing by a power of 10 makes a number (move the decimal point to the left the same # of places as the power.

47 59 Scientific Notation a special form of writing numbers. It is most useful for very large (or very small) numbers Form: Exactly one non-zero digit left of decimal point Zero or more digits right of decimal point Multiplied by power of 10 to express place value Ex Which of the following are in scientific notation? Converting Scientific Notation to Place Value Form 1. Decide if exponent will make the decimal larger or smaller (pos. or neg. exponent) 2. Move decimal point the number of places in exponent 3. Fill in zeroes if necessary Converting Place Value Form to Scientific Notation 1. Put decimal point after the first non-zero digit 2. Count how many places you moved 3. Write that number as the exponent (positive exp. for numbers > 1) Polynomials Unit Price - Price per single unit (1 in denominator)

48 Dividing Decimals; Square Roots Dividing decimals by a whole number 1. Divide as if the numerator were a whole number 2. Write the quotient above, lining up decimal points 3. Continue dividing until: a. There is no remainder OR b. The decimal repeats itself OR c. If you are told to round, you have divided one place past the place you are rounding to. Dividing decimals by a decimal 1. Write as a fraction 2. Move decimal point in both numerator & denom. until the denom. is a whole number. 3. Divide as above Converting fractions and mixed numbers to decimals 1. If mixed, write integer part to the left of the decimal point 2. Divide out the fraction by long division and write it to the right of the point

49 61 Finding Square Roots of Non-Perfect Squares irrational number can t be expressed as a/b. The decimal never repeats, never ends The best way to find a square root: 1. approximately 2. exactly Decimal Perfect Squares and Roots -- Observe the squares of some decimals: (.2) 2 =.04 1 place 2 places (.02) 2 =.0004 (.002) 2 = Conclusion: A perfect square has a(n) number of decimal places A square root has the number of places as the square. Polynomials Applications Circles circumference length of the outside of a circle (perimeter, curved). radius distance from center to edge of a circle diameter distance across a circle from edge to edge, through the center Formula: C = 2πr or C = πd

50 Order of Operations with Decimals Recall: PEMDAS Mixed Decimals and Fractions change to all decimals or to all fractions Note: Changing to all fractions is often easier If fractions change to a repeating decimal, must use fractions, e.g. 3 1 If fractions are not easy to convert (too much long division), fractions are better 5 e.g. 32 Changing to decimals is only better if the fractions are easy to see as decimals Weighted Mean - sometimes, certain categories count more heavily than others GPA (grade point average) A = 4.0, B = 3.0, C = 2.0, D = 1.0 How heavily a class is weighted depends on the number of units(credits). GPA = total grade points/# of units Ex Find the GPA of the following schedule to the nearest tenth. Class Units Grade Grade value Total grade points English A Math 20 5 C Biology B Music A

51 Ex In my class, exams are weighted 50%, homework is weighted 25%, and the final is weighted 25%. If Pat s exam avg. is 80, the final exam is 75, and homework is 40, what is her final average and grade? 63 Area formulas Triangle: A = ½ bh Trapezoid: A = ½ h(a + b) Circle: A = πr 2 Volume of a cylinder/straight-sided object V = Ah, where A = area of a slice h = height

52 Solving with Decimals 2 Methods for isolating the variable Method 1: Get rid of all decimals by moving the point the same number of places in EVERY term. Method 2: Keep all decimals until the end Translating to algebra Pythagorean Theorem legs 2 sides touching the right angle hypotenuse side opposite the right angle

53 65 Applications Ex A phone company charges $40/month with 500 free minutes. After 500 minutes, $0.50/minute is charged. If Ray s bill is $152.50, how many minutes were used? Using a table (2 types) Recall algebraic expressions for common constructions: 5 more hours than last week twice as many hot dogs as burgers 12 items all together Ex Ana has 20 quarters and dimes totaling $3.35. How many of each coin does she have? category value number amount Try Candy costs $0.75 and soda costs $1.25. If $19.50 is spent and twice as many sodas as candy bars are bought, how many of each is bought?

54 Estimation Supplement There are 2 goals of estimation: 1. To round complicated numbers so they are simple enough to do calculations in your head 2. To keep close to the original number so you have a ballpark idea of the value. Based on these 2 conditions, what would be a good estimate of $ ? 66 Estimating sums differs from estimating products, especially when one number is relatively large compared to the other number(s). Estimating Sums Compare the sum: with the number = 10.1 Adding 0.1 to 10 hardly changes the 10, so we can ignore the 0.1 when adding. When rounding, we should keep the highest 1 or 2 place values, since adding more than 2 digits becomes complicated. Estimating Sums Procedure 1. Choose the largest number and round it to the highest 1 or 2 place values. 2. Round all other numbers to those same 1 or 2 place values. Some small numbers may become zero. 3. Add Ex a Estimate the sum: House: $123, Repairs $ 21, Landscaping $ 1, Air freshener $ 2.39

55 67 Estimating Products Now compare the number 10 with the product 10 x (= 10 x 1) 10 x 0.1 But multiplying 10 by 0.1 makes it much smaller than 10 (big change). So we must keep both numbers in the calculation when doing multiplication. We should round each number to one non-zero digit, because multiplying with 2 or more digits becomes complicated. Estimating Products Procedure 1. Round each number to one non-zero digit, keeping correct place value 2. Multiply Ex b Estimate the product: ( )( )( ) Ex c Estimate the cost of 980 doses of medication if each dose is mg and the cost is $49.85/mg. (Actual cost is $301.42) Estimating Quotients Procedure 1. Round the divisor (denominator) to one non-zero digit, keeping correct place value. 2. Round the numerator to a number easily divided by the denominator. Ex d Estimate the quotient 19, Note: You can quickly check the answers to any complicated problem for ballpark values using estimates.

56 Ratios and Rates ratio a comparison of 2 quantities using a fraction (quotient) The ratio of a to b: a/b or a:b Ratios may use like units (cancel units) or unlike units (keep units) rate - a ratio comparing 2 different measurements (? Unit ratio ratio where the denominator number is 1 Ex A college has 5000 students and 250 faculty. Find the student to faculty ratio. Ex May owes $30,000 on a house, and value of the house (asset) is $120,000. Find the debt to asset ratio (this ratio is sometimes called the debt ratio ). Unit rate = denominator number is 1 Unit price cost per unit (cost/unit) Probability 1. Write the number of desired choices in numerator 2. Write the number of all possible choices in denominator

57 Proportions Proportion 2 ratios that equal each other: a c = b d Cross Multiplying (cross products) a c If =, then ad = bc b d Testing whether ratios are proportions (cross multiply) Solving Proportions 1. Cross multiply to eliminate fractions 2. Solve as before Cancelling Shortcuts 1. You can cancel straight up/down (this is the same as reducing a fraction) 2. You can cancel straight across (same as multiplying both sides of an equation) 3. You can t cancel diagonally can t cross cancel Solving proportion problems chart Quantity 1 Quantity 2 Case 1 Case 2

58 American Measurement (English Units) 12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd) 5280 feet (ft) = 1 miles ENGLISH MEASURES AND EQUIVALENTS Length Time 60 seconds (sec) = 1 minute (min) 60 minutes (min) = 1 hour (hr) 24 hours (hr) = 1 day 7 days = 1 week Liquid Volume Weight 3 teaspoons (tsp) = 1 tablespoon (Tbsp) 16 ounces (oz) = 1 pound (lb) 8 fluid ounces (fl. oz. or just oz) = 1 cup (c) 2000 pounds (lb) = 1 ton 2 cups (c) = 1 pint (pt) 2 pints (pt) = 1 quart (qt) 4 quarts (qt) = 1 gallon dimensional analysis converting from one type of unit to another using unit fractions unit fractions - fractions that equal one, formed by dividing 2 sides of an equation Conversion Procedure 1. Write equations containing the original and desired quantities 2. Write the original quantity on top 3. Multiply by unit fractions, arranging units to cancel 4. Cancel units, put numbers together

59 Metric Measurement metric system more systematic than American/English units (easier calculations) kilo hecto deka base deci centi milli METRIC LENGTH VOLUME WEIGHT 1 kilometer (km) = 1000 m 1 kiloliter (kl) = 1000 l 1 kilogram (kg) = 1000 g 1 hectometer (hm) = 100 m 1 hectoliter (hl) = 100 l 1 hectogram (hg) = 100 g 1 dekameter (dam) = 10 m 1 dekaliter (dal) = 10 l 1 dekagram (dag) = 10 g 1 meter = 1 meter 1 liter = 1 liter 1 gram = 1 gram 10 decimeter (dm) = 1 m 10 deciliter (dl) = 1 l 10 decigram (dg) = 1 g 100 centimeter (cm) = 1 m 100 centiliter (cl) = 1 l 100 centigram (cg) = 1 g 1000 millimeter (mm) = 1m 1000 milliliter (ml) = 1 l 1000 milligram (mg) = 1 g Other Common Prefixes kilobytes (KB) megabytes (MB) gigabytes (GB) terabytes (TB) milliseconds (msed) microseconds (µsed) nanoseconds (nsec) picoseconds (psec) Miscellaneous metric equivalencies 1 metric ton = 1000 kg 1 milliliter (ml) = 1 cubic centimeter (cc)

60 Converting between American (English) and Metric Systems English- Metric Conversions 1 inch 2.54 centimeters 1 foot meter 1 yard meter 1 mile kilometers 1 quart liter 1 gallon liters 1 ounce grams 1 pound grams Metric-English Conversions 1 centimeter inch 1 meter feet 1 meter yards 1 kilometer mile 1 liter quarts 1 liter gallon 1 gram ounce 1 gram pound 9 Temperature formula: F = C + 32, where F = Fahrenheit temp., C = Celsius temp. 5

61 Applications (skip mortgage ratios) Sometimes we make our own conversion equations that are not on standardized charts, based on the information given. Ex A veterinarian administers Butorphanol (a sedative) to a 1000 lb. horse. The recommended dosage is 0.02 mg/kg. A vial of the drug has a concentration of 10 mg/ml. How much of the drug should be given? Ex A certain cereal contains 6 grams of fiber per serving. The recommended intake of fiber is 24 grams per day, there are 12 servings of cereal in the box, and each box costs $6. How much does it cost for a year s worth of fiber using this cereal?

62 Introduction to Percent percent out of % = Converting percent to a fraction 1. Replace % with / Move decimal point if need (top & bottom) 3. Reduce if needed Converting percent to a decimal 1. Move decimal point 2 places to the left (smaller) 2. Remove % symbol Converting a fraction or decimal to percent 1. First, write as a decimal (if a repeating decimal, write out at least 3 places) 2. Multiply by 100% (move decimal point 2 places and tack on % symbol) (Note: 100% = no change)

63 Translating Percent Sentences of multiply out of divide is equals Method 1: Use Multiplying Equation ( of ) percent of whole is part OR part is percent of whole Example type Word phrase Equation unknown part unknown whole unknown percent Note: When solving or performing calculations, represent percent as a decimal Method 2: Use Dividing Equation ( out of ) part percent = whole, and percent = percent number 100

64 Solving Percent Problems We will make use of the relationships: part 1) part = percent of whole and 2) percent = whole = percent number 100 Finding Part Ex A restaurant tip is 15% of the bill. If $60 is spent on food, what is the tip? Finding Whole Ex A college accepts 40% of all applicants. If 500 students were accepted, how many applied? Finding percent Ex 2000 DVD s are produced, and 37 are damaged. What percent can be sold (undamaged)?

65 Solving Problems Percent Increase or Decrease Note: Both increase and decrease are a part of the original amount Increase increase = percent old new = old + increase = words to show increase increased by, up, raise(d), add, bonus, over, above, markup Decrease decrease = percent old new = old - decrease = words to show decrease - discount, sale, off, down, reduce (reduction), deduction, loss, pay cut

66 Solving Problems with Interest Principal, P amount invested Interest, I amount added to principal Interest rate, r percent used to calculate interest usually annual Simple interest interest calculated for a year using principal, rate, and time (same amount of interest is paid each year) Balance, B (most books call this Amount, A) total of principal and interest Simple Interest Formula: I = Prt Balance Formula B = P + I =

67 The Rectangular Coordinate System axis number line used to locate a point point a location in space ordered pair (x,y) the coordinates of a point (x is always first) Ex a Find the coordinates of the points shown Ex b Plot the points Quadrants - 4 regions defined by the x and y axes

68 80 Exc Which quadrant is each point in? (3, -2) (-10, - 20) (4, 0) (0, -7) (-52, 37) Midpoint the point halfway between 2 points on a line segment For each coordinate, find the average of the 2 values: x-coordinate average: y-coordinate average:

69 Graphing Linear Equations (2 variables) Linear Equation - each variable term has degree 1 x 2 4x + 7 = 0 x + 8 = 12 x y = 5 A solution is a number that gives a true equation. For x + 8 = 12, the solution looks like: For x y = 5, a solution must have Finding a Solution (Procedure) An equation with 2 variables may have many solutions. To find one: 1. Choose any number for either x or y. 2. Plug that value into the equation to calculate the number for the other variable. Write as an ordered pair. The solutions of a linear equation can be graphed as points. It takes at least points to determine a line Graphing a Linear Equation (procedure) 1. Find 2 (or more) solutions to the equation 2. Plot the solutions as points 3. Draw the line through the points

70 82 Vertical and Horizontal Lines x and y intercepts x-intercept: point where the line crosses the x-axis (y = 0) y-intercept: point where the line crosses the y-axis (x = 0) To find the intercepts: 1. Set opposite coordinate = 0 2. Solve for the desired coordinate 3. Write ordered pair with both coordinates 3 exceptions 1. Vertical Line 2. Horizontal Line 3. Line through origin

71 83 Slope Supplement A linear equation can be written several ways including: 1. General Form: Ax + By = C, for example 3x + y = 2 2. Slope-Intercept Form: y = mx + b, for example y = -3x + 2 Note that the equations 3x + y = 2 and y = -3x + 2 are different forms of the same equation. For y = mx + b, m = slope b = y-intercept (y coordinate only) What is slope? Generically speaking, it's the slant or steepness of an object, like a hill, roof, or line. In the linear equation above, the symbol "m" is used to represent slope for a line, and the value depends on the direction and amount of slant. If m is positive, the line from left to right If m is negative, the line from left to right If m has a large absolute value, the line rises or falls If m has a small absolute value, the line rises or falls Ex a We can also get a general sense of the location of a line by looking at "b" If b is positive, the line intersects the y-axis in the positive region, above the x- axis If b is negative, the line intersects the y-axis in the negative region, below the x- axis Ex b

72 84 Slope as a ratio rise vertical change Slope is sometimes defined as =. run horizontal change Other words which mean the same thing as slope: grade (of a road) pitch (of a roof) Ex c What percent grade does a hill have if it rises 300 feet over a distance of 5000 feet? Ex d If the pitch of a roof is 25%, what is the distance from the horizontal support to the peak if the horizontal support spans 80 feet all the way across? Slope formula y 2 - y1 m =, where (x 1, y 1 ) and (x 2, y 2 ) are the coordinates of a point. x - x 2 1 Ex e Find the slope of the line through the points (4, -1) and (-3, 0), and graph the line.

73 85 Ex f Find the slope of the line through the points (-3, 1) and (2,1), and graph the line. Ex g Find the slope of the line through the points (-2, 7) and (-2, 4), and graph the line. Finding Slope from an Equation Procedure: 1. Isolate y on the left. The equation now looks like y = mx + b. 2. The coefficient of x is the slope 1 Ex h Find the slope of y = x Ex i Find the slope of 2x 3y = 6

74 86 Slope Supplement Exercises (do all) For # 1-6, tell whether a line with the given slope will be rising, falling, horizontal, or vertical. 1. m = m = 0 3. m = -2/3 4. m = ¼ 5. m = undefined 6. m = 7 For each of the following lines (y = mx + b) tell whether a) m is positive or negative and b) whether b is positive or negative. 11. If a road has an 8% grade up a hill, and the horizontal run is 2000 feet from the edge of the hill to the center, how high is the peak? 12. If skater's ramp has the following dimensions, find the slope in percent: 13. If a roof has the following dimensions, find the slope in percent: 14. Find the slope of the line through the points (-7, 4) and (2, 4). 15. Find the slope of the line through the points (0, 6) and (2, 3). 16. Find the slope of the line through the points (1, -3) and (1, -1). 17. Find the slope of the line through the points (5, 1) and (3, -3). 18. Find the slope of the line y = -5x Find the slope of the line 3x y = Find the slope of the line 2x + 5y = 9

75 87 12 inches (in) = 1 foot (ft) 3 feet (ft) = 1 yard (yd) 5280 feet (ft) = 1 miles Conversion Charts for Exams ENGLISH MEASURES AND EQUIVALENTS Length Time 60 seconds (sec) = 1 minute (min) 60 minutes (min) = 1 hour (hr) 24 hours (hr) = 1 day 7 days = 1 week Liquid Volume 3 teaspoons (tsp) = 1 tablespoon (Tbsp) 8 fluid ounces (fl. oz. or just oz) = 1 cup (c) 2 cups (c) = 1 pint (pt) 2 pints (pt) = 1 quart (qt) 4 quarts (qt) = 1 gallon Weight 16 ounces (oz) = 1 pound (lb) 2000 pounds (lb) = 1 ton kilo hecto deka base deci centi milli METRIC LENGTH VOLUME WEIGHT 1 kilometer (km) = 1000 m 1 kiloliter (kl) = 1000 l 1 kilogram (kg) = 1000 g 1 hectometer (hm) = 100 m 1 hectoliter (hl) = 100 l 1 hectogram (hg) = 100 g 1 dekameter (dam) = 10 m 1 dekaliter (dal) = 10 l 1 dekagram (dag) = 10 g 1 meter = 1 meter 1 liter = 1 liter 1 gram = 1 gram 10 decimeter (dm) = 1 m 10 deciliter (dl) = 1 l 10 decigram (dg) = 1 g 100 centimeter (cm) = 1 m 100 centiliter (cl) = 1 l 100 centigram (cg) = 1 g 1000 millimeter (mm) = 1m 1000 milliliter (ml) = 1 l 1000 milligram (mg) = 1 g English- Metric Conversions 1 inch 2.54 centimeters 1 foot meter 1 yard meter 1 mile kilometers 1 quart liter 1 gallon liters 1 ounce grams 1 pound grams Metric-English Conversions 1 centimeter inch 1 meter feet 1 meter yards 1 kilometer mile 1 liter quarts 1 liter gallon 1 gram ounce 1 gram pound Misc. Formulas 1 metric ton = 1000 kg 1 milliliter (ml) = 1 cubic centimeter (cc) 9 F = C + 32, where F = Fahrenheit temp., C = Celsius temp. 5